Visualisation of Conformal Mappings

Wouter M. Koolen

2017-03-20

Introduction

I started studying conformal mappings some time ago, and made this visualisation that I would like to share. A conformal mapping locally preserves angles (i.e. its Jacobian is a scaled rotation matrix). In the figure below the black regular grid is mapped to the yellow warped grid. As you can see, all intersections of yellow lines are still at right angles (as they were in the black grid), even though the lines are themselves bent.

Conformal mapping example. We start with the black regular grid as the object to be mapped. The pink grid is obtained by a translation (which is conformal). The green grid is then obtained by inversion of the pink grid in the red circle (spherical inversion is conformal). The blue grid is again obtained by translating the green grid. And finally, we invert the blue grid in the red circle to obtain the yellow grid.

Conformal mappings are not linear, but they can be given a linear representation by embedding in Minkowski space with two additional dimensions. (This is similar to how homogeneous coordinates make translation and projection linear by increasing the dimension by one.) I recommend (Dorst, Fontijne, and Mann 2009) for more details.